Adiabatic invariance, in which certain quantities of a system remain unchanged as a parameter of the system is infinitely slowly altered, plays a fundamental role in many areas of physics. For any harmonic oscillator, the adiabatic invariant is the energy divided by the frequency. When the alterations are slow but occur over a finite time, there is predicted to be an exponential suppression of the change in adiabatic invariant; that is, if epsilon is a dimensionless positive number that tends to zero in the limit of infinitely slow alterations, then the change in adiabatic invariant is proportional to exp(-1/ epsilon). We report numerical simulations of three oscillators whose parameters are varied at rates ranging from very slow to very fast compared to the oscillation frequency. The models are single-degree-of-freedom oscillators that are based on simple physical systems. The exponential suppression is not observed, which indicates that its observation may be extremely difficult or impossible. Furthermore, the change in adiabatic invariant is found to depend upon the initial phase even in the limit of infinitely slow changes. In the case of abrupt alterations, the numerical simulations verify some theoretical calculations, but reveal that other theoretical calculations are incorrect.

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